hdmap1l6i - Mathbox for Norm Megill

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Theorem hdmap1l6i 39829

Description: Lemmma for hdmap1l6 39832. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.)

**Hypotheses**
Ref	Expression
hdmap1l6.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1l6.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1l6.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1l6.p	⊢ + = (+_g‘𝑈)
hdmap1l6.s	⊢ − = (-_g‘𝑈)
hdmap1l6c.o	⊢ 0 = (0_g‘𝑈)
hdmap1l6.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1l6.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1l6.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1l6.a	⊢ ✚ = (+_g‘𝐶)
hdmap1l6.r	⊢ 𝑅 = (-_g‘𝐶)
hdmap1l6.q	⊢ 𝑄 = (0_g‘𝐶)
hdmap1l6.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap1l6.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1l6.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1l6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1l6.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1l6cl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1l6.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
hdmap1l6i.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
hdmap1l6i.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1l6i.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
hdmap1l6i.yz	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))

**Assertion**
Ref	Expression
hdmap1l6i	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Proof of Theorem hdmap1l6i Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1l6.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1l6.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
5		hdmap1l6.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		hdmap1l6cl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
7	6	eldifad 3900	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
8		hdmap1l6i.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3900	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10	1, 2, 3, 4, 5, 7, 9	dvh3dim 39457	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
11		hdmap1l6.p	. . . 4 ⊢ + = (+_g‘𝑈)
12		hdmap1l6.s	. . . 4 ⊢ − = (-_g‘𝑈)
13		hdmap1l6c.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
14		hdmap1l6.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
15		hdmap1l6.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
16		hdmap1l6.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
17		hdmap1l6.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
18		hdmap1l6.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
19		hdmap1l6.l	. . . 4 ⊢ 𝐿 = (LSpan‘𝐶)
20		hdmap1l6.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
21		hdmap1l6.i	. . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
22	5	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23		hdmap1l6.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
24	23	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷)
25	6	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
26		hdmap1l6.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
27	26	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
28		hdmap1l6i.xn	. . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
29	28	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
30		hdmap1l6i.yz	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
31	30	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
32	8	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
33		hdmap1l6i.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
34	33	3ad2ant1 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑍 ∈ (𝑉 ∖ { 0 }))
35		eqid 2738	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
36	1, 2, 5	dvhlmod 39121	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	36	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod)
38	3, 35, 4, 36, 7, 9	lspprcl 20238	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
39	38	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
40		simp2 1136	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉)
41		simp3 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
42	13, 35, 37, 39, 40, 41	lssneln0 20212	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 }))
43	1, 2, 3, 11, 12, 13, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 32, 34, 42, 41	hdmap1l6h 39828	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
44	43	rexlimdv3a 3214	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))))
45	10, 44	mpd 15	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 ∖ cdif 3885 {csn 4563 {cpr 4565 ⟨cotp 4571 ‘cfv 6435 (class class class)co 7277 Basecbs 16910 +_gcplusg 16960 0_gc0g 17148 -_gcsg 18577 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 HLchlt 37361 LHypclh 37995 DVecHcdvh 39089 LCDualclcd 39597 mapdcmpd 39635 HDMap1chdma1 39802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-riotaBAD 36964

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-ot 4572 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8040 df-undef 8087 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-map 8615 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-n0 12232 df-z 12318 df-uz 12581 df-fz 13238 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-mre 17293 df-mrc 17294 df-acs 17296 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36987 df-lshyp 36988 df-lcv 37030 df-lfl 37069 df-lkr 37097 df-ldual 37135 df-oposet 37187 df-ol 37189 df-oml 37190 df-covers 37277 df-ats 37278 df-atl 37309 df-cvlat 37333 df-hlat 37362 df-llines 37509 df-lplanes 37510 df-lvols 37511 df-lines 37512 df-psubsp 37514 df-pmap 37515 df-padd 37807 df-lhyp 37999 df-laut 38000 df-ldil 38115 df-ltrn 38116 df-trl 38170 df-tgrp 38754 df-tendo 38766 df-edring 38768 df-dveca 39014 df-disoa 39040 df-dvech 39090 df-dib 39150 df-dic 39184 df-dih 39240 df-doch 39359 df-djh 39406 df-lcdual 39598 df-mapd 39636 df-hdmap1 39804

This theorem is referenced by: hdmap1l6j 39830

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